No, half a gal is ok for a quarantine or treatment tank! Just remember the daily water changes. The salt can build up, and this is harmful. Premixing the water in a separate container is perfect!
The pictures aren't the clearest, but if he's like my boy he won't sit still for photos. I believe you do have a case of fin chewing going on. IF the ends look kinda feathery, that's a dead give away.
This thread has some really good pictures of tail biting.
IT looks an awful lot like the bad pictures I took of my own boy after fin chewing. It also looks like he might have gotten at his bottom fin as well as both of his tails.
Right now you want to find out why he bit his tail. There are a few theories, one is boredom. Does he have plants (live, silk, any?) a hidey hole? a cave or mug or even just a colourful statue with a hole? OR has he been stressed out? Have you made any changes? new kind of food, or water conditioner, have you changed anything on the inside OR outside of the tank? Yes, some bettas can be spazzy enough that changing something outside of their tank can make them chew their tail.
The other question is does he have any neighbours? Another fish or betta he can see? Something that could be mistaken for a colourful betta? Does he flare at the sides of his tank much? Or is he quiet and sulky?
Alot of questions I know, but there are so many reasons. There is a third, which is truly the worst. He does it purposefully, as in he finds his fins to heavy, or he feels the need for a "trim" so to speak. There is another theory that they do it to remove rotting or infected fins, but I have no experience with that one.
PHEW! Sorry for the real book, after discovering my own boy having absolutely mangled his tail, and I knew it was stress induced and that it was him, I started to really read up on the condition. I was in tears after that first time, so I understand your concern.
Glad to hear about the live mosquitoes. I'm just too squeamish to go that far, besides, they would over run the backyard, and front yard, and uh, yea. Once you start to see new growth, this generally is clear or sort of whitish in colour it's probably safe to stop with the salt treatment. This can take 4-5 days, or it can take 2-3 weeks. Since you said the ends don't really look black, the salt is more a preventative measure, to stop infection and or fin rot. Sadly once they start biting, it can be a chronic condition.
OH! if you do believe it is boredom, I read someone having the idea of say a ping pong ball or bottle cap floating in his tank, it might give him something to play with, or at least take out some aggressions. I plan on trying this when I get home to my boy. As if the floating plant for a bed wasn't enough!